ON INFINITE CHAINS OF INTERMEDIATE FIELDS

Year 2012, Volume: 11 Issue: 11, 165 - 176, 01.06.2012

David E. Dobbs

Abstract

If K ⊂ L is an infinite-dimensional algebraic field extension and
the (infinite) cardinal number ℵ := [L : K] (the K-vector space dimension of
L), then there exists an infinite maximal chain, C, consisting of fields contained
between K and L, such that the cardinality of C is at most ℵ. If K ⊂ L is a
J-extension, then every maximal chain of intermediate fields has cardinality
ℵ0. However, an example is given where K ⊂ L has maximal chains, D and
E, of intermediate fields such that the cardinalities of D and E are ℵ and 2ℵ,
respectively.

Keywords

field extension, intermediate field, maximal chain, infinite cardinal number, algebraic extension, transfinite induction, J-extension, transcendence degree, integral domain, classical D + M construction